Apparatus and method for stiction compensation in a process control system

ABSTRACT

An apparatus, method, and computer program for stiction compensation in a process control system are provided. A determination is made as to whether a valve is suffering from stiction. A control signal provided to the valve is adjusted in order to at least partially compensate for the stiction suffered by the valve. Adjusting the control signal could include (i) adjusting the control signal to cause the valve to move into a steady-state position and (ii) adjusting the control signal to cause the valve to remain in the steady-state position. Adjusting the control signal to cause the valve to move into the steady-state position could include (a) adjusting the control signal to cause the valve to move from a current position into a new position and (b) adjusting the control signal to cause the valve to move from the new position into the steady-state position.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 U.S.C. §119(e) to U.S.Provisional Patent Application No. 60/725,668 filed on Oct. 13, 2005,which is hereby incorporated by reference.

TECHNICAL FIELD

This disclosure relates generally to control systems and morespecifically to an apparatus and method for stiction compensation in aprocess control system.

BACKGROUND

Processing facilities are typically managed using process controlsystems. Among other functions, these control systems often manage theuse of valves in the processing facilities. The valves typically controlthe flow of materials in the facilities. Example processing facilitiesinclude manufacturing plants, chemical plants, crude oil refineries, andore processing plants. In these facilities, the valves may control theflow of water, oil, hydrochloric acid, or any other or additionalmaterials in the facilities.

The valves used in the processing facilities often suffer from a numberof problems or defects. For example, a valve may suffer from valve“stiction.” Valve stiction, which is short for static friction, refersto the valve's resistance to the start of motion. It occurs when thevalve fails to respond to pressure meant to adjust the opening of thevalve. The valve fails to respond until additional pressure is added,which then causes the valve to open or close more than desired. Thisprocess is then repeated, generally causing an oscillation in the valve.This often limits or prevents the control systems from accuratelycontrolling the flow of materials using the valve. It also often wastesenergy and causes excessive wear on the valves.

SUMMARY

This disclosure provides an apparatus and method for stictioncompensation in a process control system.

In a first embodiment, an apparatus includes at least one memoryoperable to store operating data associated with a valve. The apparatusalso includes at least one processor operable to identify whether thevalve is suffering from stiction using at least part of the operatingdata. The at least one processor is also operable to adjust a controlsignal provided to the valve in order to at least partially compensatefor the stiction suffered by the valve.

In particular embodiments, the at least one processor is operable toadjust the control signal by (i) adjusting the control signal to causethe valve to move into a steady-state position and (ii) adjusting thecontrol signal to cause the valve to remain in the steady-stateposition.

In other particular embodiments, the at least one processor is operableto adjust the control signal to cause the valve to move into thesteady-state position by (a) adjusting the control signal to cause thevalve to move from a current position into a new position and (b)adjusting the control signal to cause the valve to move from the newposition into the steady-state position.

In a second embodiment, a method includes identifying whether a valve issuffering from stiction. The method also includes adjusting a controlsignal provided to the valve in order to at least partially compensatefor the stiction suffered by the valve.

In a third embodiment, a computer program is embodied on a computerreadable medium and is operable to be executed by a processor. Thecomputer program includes computer readable program code for identifyingwhether a valve is suffering from stiction. The computer program alsoincludes computer readable program code for adjusting a control signalprovided to the valve in order to at least partially compensate for thestiction suffered by the valve.

Other technical features may be readily apparent to one skilled in theart from the following figures, descriptions, and claims.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of this disclosure, reference is nowmade to the following description, taken in conjunction with theaccompanying drawings, in which:

FIG. 1 illustrates an example process control system including a valve;

FIG. 2 illustrates an example valve in a process control system;

FIG. 3 illustrates example fluctuations associated with a valvesuffering from stiction in a process control system;

FIG. 4 illustrates an example model of a process control system;

FIG. 5 illustrates an example method for stiction compensation in aprocess control system;

FIGS. 6 through 8 illustrate example stiction compensation in a processcontrol system;

FIGS. 9 through 10B illustrate an example technique for qualitativelydetermining an amount of stiction in a valve; and

FIGS. 11A and 11B illustrate an example technique for quantitativelydetermining an amount of stiction in a valve.

DETAILED DESCRIPTION

FIG. 1 illustrates an example process control system 100 including avalve 102. The process control system 100 shown in FIG. 1 is forillustration only. Other systems may be used without departing from thescope of this disclosure.

In the illustrated example, one or more materials flow through a pipe104, and the flow of materials through the pipe 104 is controlled by thevalve 102. The pipe 104 represents any suitable structure capable offacilitating the transport of one or more materials. The pipe 104 could,for example, represent a steel or plastic pipe or tube capable offacilitating the transport of oil, water, hydrochloric acid, or anyother material or materials.

The valve 102 controls the rate at which the material or materials flowthrough the pipe 104. The valve 102 may, for example, change an openingin the pipe 104, where a larger valve opening allows more material toflow through the pipe 104. The valve 102 includes any structure capableof controlling the flow of one or more materials through a pipe 104 orother structure. For example, the valve 102 may include a stem capableof moving a plug to change the size of an opening. One exampleembodiment of a valve 102 is shown in FIG. 2, which is described below.

In the illustrated example, the process control system 100 also includesa sensor 106, a controller 108, and a valve actuator/positioner 110. Thesensor 106 monitors one or more characteristics associated with theprocess control system 100. For example, the sensor 106 may measure theflow rate of a material flowing through the pipe 104. The sensor 106could monitor any other or additional characteristics of the materialflowing through the pipe 104 or of the process control system 100, suchas pressure or temperature. The sensor 106 also transmits or outputs asignal 112 to the controller 108, where the signal 112 includes valuesidentifying the measurements made by the sensor 106. The flow rate orother monitored characteristic may be referred to as a process variable,and the signal 112 provided to the controller 108 may be referred to asa process variable (PV) signal. The sensor 106 includes any hardware,software, firmware, or combination thereof capable of measuring at leastone characteristic of the process control system 100.

The controller 108 controls the opening and the closing of the valve 102in the system 100. In this example embodiment, the controller 108 usesthe process variable signal 112 provided by the sensor 106 and asetpoint (SP) signal 114 to control the valve 102. The setpoint signal114 identifies the desired value for the process variable signal 112,and the setpoint signal 114 may or may not vary over time. As aparticular example, the controller 108 may adjust the valve opening sothat the flow rate of material through the pipe 104 remains at or near alevel indicated by the setpoint signal 114. Using the process variablesignal 112 and the setpoint signal 114, the controller 108 generates andtransmits an output signal (OP) 116. The output signal 116 controls theopening and closing of the valve 102. Ideally, the output signal 116controls the valve 102 so that the process variable signal 112 remainsat or near the setpoint signal 114. The controller 108 includes anyhardware, software, firmware, or combination thereof for controlling theoperation of the valve 102. As a particular example, the controller 108could include at least one processor 118 and at least one memory 120capable of storing data and instructions (such as one or more softwareroutines) executed by the at least one processor 118.

The valve actuator/positioner 110 opens and closes the valve 102 basedon the output signal 116. In general, an actuator may represent adiaphragm or other structure that physically causes the stem of thevalve 102 to move (thereby moving the plug of the valve 102), while apositioner controls the valve stem so that it corresponds to the outputsignal 116. Valves 102 often include actuators and may or may notinclude positioners. The valve actuator/positioner 110 includes anystructure(s) capable of opening and/or closing a valve 102.

In one aspect of operation, the process control system 100 includes astiction compensator 122. The stiction compensator 122 helps tocompensate for stiction in the valve 102. Stiction occurs when the valve102 fails to respond to pressure from the valve actuator/positioner 110until additional pressure is applied to the valve 102. At that point,the valve 102 jumps to a larger or smaller opening than desired, and theprocess is then repeated. Stiction typically causes persistentoscillations in the position of the valve 102, meaning the actualposition of the valve 102 typically swings back and forth around adesired or steady-state position (defined by setpoint signal 114).

The stiction compensator 122 helps to reduce the effects of valvestiction in the process control system 100. For example, the stictioncompensator 122 may generate a compensation signal 124, whichcompensates the output signal 116 generated by the controller 108 tohelp force the valve 102 into a steady-state position. Once the valve'ssteady-state position is reached, the stiction compensator 122 maycompensate the output signal 116 (via the compensation signal 124) tokeep the controller 108 from moving the valve 102 away from itssteady-state position. In this way, the stiction compensator 122 mayhelp to reduce or prevent valve stiction from significantly affectingthe overall operation of the process control system 100.

The stiction compensator 122 may include any hardware, software,firmware, or combination thereof for at least partially compensating forstiction in one or more valves 102. For example, the stictioncompensator 122 could include at least one processor 126 and at leastone memory 128 capable of storing data and instructions (such as one ormore software routines) executed by the at least one processor 126. Asparticular examples, the stiction compensator 122 could represent anapplication or other logic executed by a stand-alone unit or anothercomponent in the process control system 100 (such as the controller108). The stiction compensator 122 could also represent a hardware orother module implemented in a stand-alone unit or another component inthe process control system 100. The stiction compensator 122 could beimplemented in any other suitable manner.

FIG. 2 illustrates an example valve 102 in a process control system. Theembodiment of the valve 102 shown in FIG. 2 is for illustration only.Other embodiments of the valve may be used in the process control system100 without departing from the scope of this disclosure.

As shown in this example, the valve 102 generally includes a stem 202and a plug 204. The stem 202 can be moved up and down to move the plug204, which changes the amount of material flowing through an opening 206in the valve 102. The valve actuator/positioner 110 generally includes aplate 208 and a spring 210. A control signal (such as a pneumatic airsignal or other output signal 116) can be increased to push down againstthe plate 208, which pushes the plug 204 towards the opening 206. Thecontrol signal can also be decreased to allow the spring 210 to push upagainst the plate 208, which pulls the plug 204 away from the opening206. In this way, the amount of material flowing through the valve 102can be controlled.

Stiction may cause the plug 204 to move more than desired when the valve102 is being adjusted. Stiction can occur for a wide range of reasons,including seal degradation, lubricant depletion, foreign matterintrusion, activation at metal sliding surfaces at high temperatures,and packing around the stem 202 in the valve 102. Stiction often variesover time and can lead to non-uniform wear of the valve 102.

When the controller 108 attempts to adjust the valve 102 so that theprocess variable signal 112 is at or near the setpoint signal 114,stiction often causes persistent oscillations in the position of thevalve 102. For example, the controller 108 may attempt to slow the flowof material through the valve 102, so the controller 108 beginsincreasing the value of the output signal 116. However, staticfrictional forces prevent the valve 102 from moving, so the controller108 continues to increase the value of the output signal 116. At somepoint, the static frictional forces in the valve 102 are overcome, butthe excessive output signal 116 causes the valve 102 to close more thandesired. The controller 108 then begins attempting to move the valve 102in the opposite direction. A similar process can then occur, wherestiction causes the valve 102 to open more than desired.

This behavior is shown in FIG. 3, which illustrates example fluctuationsassociated with a valve 102 suffering from stiction in a process controlsystem. In FIG. 3, the process variable signal 112, the setpoint signal114, and the output signal 116 are charted over time. As shown here, theprocess variable signal 112 generally oscillates between time periodswith higher values and time periods with lower values. This indicatesthat the position of the valve 102 is oscillating around itssteady-state position (as defined by the setpoint signal 114 used by thecontroller 108).

Also, as shown in FIG. 3, the output signal 116 generally decreasesduring the time periods when the process variable signal 112 is higherand generally increases during the time periods when the processvariable signal 112 is lower. This indicates that an increase ordecrease in the output signal 116 is typically followed by an excessivemovement of the valve 102, causing a jump in the process variable signal112. This results in oscillations in the process variable signal 112.

These oscillations often lead to increased energy consumption, wastageof materials, and less uniform end products. Reducing or eliminatingthese oscillations could yield significant commercial or economicbenefits, particularly given the large number of valves in the processindustry (currently three million or more). Even a one percent reductionin energy consumption in the process industry could result insignificant savings (such as $300 million or more per year). Also,compensating for stiction, as opposed to identifying stiction andperforming valve maintenance, could reduce the cost of maintaining thevalves, where valve maintenance could cost $400-$2,000 or more pervalve.

In general, the stiction compensator 122 may compensate or adjust theoutput signal 116 provided by the controller 108 to help reduce theeffects of stiction in the valve 102. In the absence of any setpointchanges, this may help to force the valve 102 into a steady-stateposition more quickly and remain in that steady-state position. The timerequired to reach the steady-state position may depend on the dynamicsof the particular system and the design of the stiction compensator 122.To compensate or adjust the output signal 116 provided by the controller108, the stiction compensator 122 may function to push the stem 202/plug204 of the valve 102 to its steady-state position in a desired amount oftime, such as a specified number of time blocks or steps. Once at thesteady-state position, the stiction compensator 122 may function toprevent the stem 202/plug 204 of the valve 102 from moving away fromthis steady-state position, allowing the valve position to remainundisturbed.

The following describes one particular implementation of the stictioncompensator 122 for compensating for stiction in a valve 102. Thisdescription of the stiction compensator 122 is for illustration andexplanation only. Other embodiments of the stiction compensator 122could be used without departing from the scope of this disclosure.

In discussing the operation of the stiction compensator 122, the processcontrol system 100 shown in FIG. 1 (or any other process system) couldbe represented as shown in FIG. 4. In this example, the process variablesignal (y) 112 is subtracted from the setpoint signal (y_(SP)) 114 toproduce an error signal e. The error signal e is used by a controllermodel (G_(C)) 402 to generate the output signal (m) 116. The stictioncompensator 122 produces a compensation signal (f_(k)) 124, whichadjusts the output signal 116 to produce a control signal u. The controlsignal u is provided to the valve 102.

The valve 102 is represented, in part, by a stiction non-linearity 404.Stiction causes the valve 102 to have a non-linear response to thesignal u, and that non-linear response is represented by thenon-linearity 404. Here, x represents the expected or predicted positionof the stem 202/plug 204 in the valve 102. Also, a process model orplant (G_(P)) 406 represents the linear behavior of the process system100 (including the valve 102). This means that the plant 406 representsthe valve dynamics after the start of stem/plug movement in the valve102 (when the valve position responds linearly to the output signal116), and the behavior caused by stiction is represented by thenon-linearity 404. The output of the plant 406 represents the processvariable signal 112.

Using this model as a reference, some embodiments of the stictioncompensator 122 may operate as shown in FIG. 5, which illustrates anexample method 500 for stiction compensation in a process controlsystem. As shown in FIG. 5, the stiction compensator 122 may identify aseverity of any stiction in a valve 102 at step 502. In particularembodiments, the stiction compensator 122 may identify two stictionseverity measures using different techniques. One stiction severitymeasure may be qualitative, and the other stiction severity measure maybe quantitative.

For the qualitative stiction severity measure, a qualitative patternrecognition approach can be used for stiction diagnosis. As shown inFIG. 3, stiction can create very distinctive shapes or patterns in theoutput signal 116, process variable signal 112, and other signals.Example shapes include square/rectangular, triangular, and saw-toothedshapes, depending on the type of controller 108 used. In contrast,stiction-free valves 102 often exhibit more sinusoidal oscillations inthe output signal 116 and the process variable signal 112. Patternrecognition can be used to identify differences in shape and determineif, and to what extent, stiction is present in a valve 102. Additionaldetails regarding this approach are shown in FIGS. 9 through 10B, whichare described below.

For the quantitative stiction severity measure, a Hammerstein modelidentification procedure can be used to estimate the stiction parameterin the following stiction model:

$\begin{matrix}{x_{t} = \left\{ {\begin{matrix}x_{t - 1} & {{{if}\mspace{14mu}{{u_{t} - x_{t - 1}}}} \leq d} \\u_{t} & {otherwise}\end{matrix}.} \right.} & (1)\end{matrix}$Here, x_(t) and x_(t−1) represent the present and past movements of thestem 202/plug 204 in the valve 102, u_(t) represents the present outputof the controller 108, and d represents a valve stiction band orthreshold. The Hammerstein model is shown in FIG. 4, where thenon-linearity 404 of the valve 102 caused by stiction is representedseparately from the plant 406, which includes the linear behavior of thevalve 102. Additional details regarding this approach are shown in FIGS.11A and 11B, which are described below.

Assuming stiction is present, the stiction compensator 122 thendetermines an amount of compensation necessary to move the valve 102from its current position to a steady-state position at step 504. Thestiction compensator 122 also generates and provides one or morecompensation signals (f_(k)) 124 to provide the necessary compensationat step 506. The stiction compensator 122 determines whether the valve102 has reached its steady-state position at step 508 and, if not,repeats steps 504-506.

In some embodiments, the stiction compensator 122 may perform thesesteps as follows. Assume y_(t), e_(t), m_(t), and x_(t) represent thecurrent values of the process variable signal 112, the error e, theoutput signal 116, and the stem/plug position of the valve 102. FromEquation (1), in the absence of compensation (f_(k)=0), the followingmay be true:u _(t) =m _(t) and |u _(t) −x _(ss) |≦d  (2)where x_(ss) represents the steady-state position of the valve 102. Theaddition of the compensation signal 124 at a current time t may not movethe valve 102 from its current position x_(t) to the steady-stateposition x_(ss) in a single move, and more than one move may be designedto move the valve 102 from its current position x_(t) to thesteady-state position x_(ss).

In particular embodiments, the stiction compensator 122 may design a setof compensating valve moves that push the stem 202/plug 204 of the valve102 to or near its steady-state position. Let (f_(k))_(t) represent thecompensation signal 124 that is added to the output signal (m_(t)) 116at time t. As shown in Equation (1), an amplitude change that is equalto or greater than d from its past value u_(t−1) causes the stem202/plug 204 of the valve 102 to move from its present position to a newposition. The first compensating move of the valve 102 could thereforebe designed as (f_(k))_(t)=|m_(t)|+d, such that u_(t)=m_(t)±(f_(k))_(t).Both the addition and subtraction of (f_(k))_(t) from m_(t) can causethe valve position to move such that |x_(t)|>d. The determination ofwhether to perform an addition or subtraction may depend on severalfactors, such as the proximity of the current valve operating point toits allowable or saturation limits.

At this point (current time t), |x_(t)|>d. The next compensating movecan be designed to move the valve 102 into its steady-state position.Using models of the process and the controller, the next output of thecontroller 108 can be predicted by the stiction compensator 122. Forexample, the following could be used to predict the controller output:(y _(pred))_(t+1) =G _(p) x _(t)  (3)e _(t+1)=−(y _(pred))_(t+1)  (4)m_(t+1)G_(c)e_(t)  . (5)Here, (y_(pred))_(t+1) represents the predicted controller output attime t+1. For the valve 102 to move into its steady-state position, thefollowing may be used:x _(t+1)=0→u _(t+1)=0  (6)u _(t+1)=0→m _(t+1)+(f _(k))_(t+1)=0  (7)(f _(k))_(t+1) =−m _(t+1).  (8)To summarize, two compensating moves (f_(k))_(t) and (f_(k))_(t+1) canbe made by the stiction compensator 122 to move the valve 102 into itssteady-state position, and provide stiction compensation:(f _(k))_(t) =|m _(t) |+d and u _(t) =m _(t)±(f_(k))_(t)  (9)(f _(k))_(t+1) =−m _(t+1)  . (10)

As shown in Equation (10), the second compensating move may not bedependent on the first move. Also, since the first compensating move inEquation (9) can involve either adding or subtracting a value from theoutput signal 116, there are at least two ways to design the firstcompensating move. Since controllers often have an operating rangenormalized between 0% and 100%, the first compensating move may bedesigned so that the first move is in the range 0% to 100% (the valve102 is not pushed either fully opened or fully closed). Further, whileshown here as involving two compensating moves, the movement of thevalve 102 into its steady-state position could involve any number ofcompensating moves. Using more than two compensating moves may not beoptimum with respect to the number of times that the valve 102 is moved,but it may help the process variable signal 112 reach its setpointsignal 114 more quickly.

In addition, the first compensating move could involve any signal thatmoves the valve 102 from its stuck position. Therefore, a set ofpossible first compensating moves could be represented as(f_(k))_(t)=|m_(t)+αd, where α represents a real value greater than one.This implies that there may be several ways to design the firstcompensating move, as long as the move does not cause the valve 102 tosaturate. A high magnitude compensation signal 124 could possibly beused to bring the process variable signal 112 to the setpoint signal 114relatively quickly.

Returning to FIG. 5, after the valve 102 reaches its steady-stateposition, the stiction compensator 122 takes steps to help maintain thevalve 102 in its steady-state position. The stiction compensator 122determines the amount of compensation necessary to maintain the valve102 in its steady-state position at step 510. The stiction compensator122 generates and provides one or more compensation signals (f_(k)) 124to provide the necessary compensation at step 512. If no setpoint changeis detected at step 514, the stiction compensator 122 may continue torepeat steps 510-512 to maintain the valve 102 in its steady-stateposition. Otherwise, the stiction compensator 122 may return to step 504to help move the valve 102 into a new steady-state position associatedwith a new value of the setpoint signal 114.

In particular embodiments, the stiction compensator 122 may performthese steps as follows. The compensation signal 124 can be designed tosatisfy the following condition:|m _(t)+(f _(k))_(t) |<d  (11)for all t. Under this condition, the stem 202/plug 204 of the valve 102may be maintained in its steady-state position, and the process variablesignal 112 may reach and maintain the setpoint signal 114. Also, thecompensation signal 124 could be designed so as to be constantly boundedwith a value as specified below:|(f _(k))_(t) |<d+A  (12)where A represents the amplitude of the limit cycle at the controlleroutput m.

Note that the above description of FIG. 5 has contained specificexamples of calculations, equations, and assumptions. This has been forillustration and explanation only. Other embodiments of the stictioncompensator 122 could be used that operate in any other suitable manner.

Using the technique described above with respect to FIG. 5, exampleoperations of the stiction compensator 122 are shown in FIGS. 6 through8. With respect to FIGS. 6 and 7, consider a closed-loop process where:

$\begin{matrix}{{G_{p}(z)} = \frac{{0.0091z^{- 1}} + {0.0082z^{- 2}}}{1 - {1.724z^{- 1}} + {0.748z^{- 1}}}} & (13) \\{{G_{c}(z)} = {\frac{16 - {27.53z^{- 1}} + {11.81z^{- 2}}}{1 - z^{- 1}}.}} & (14)\end{matrix}$Assume the system is sampled once per second, and the loop oscillatesdue to stiction simulated using Equation (1) with d=0.5. A two-stepcompensation (as described above with respect to steps 504-506 of FIG.5) is activated at time t=120 seconds.

As shown in FIG. 6, line 602 represents the process variable signal 112,line 604 represents the setpoint signal 114, and line 606 represents theoutput signal 116. In this simulation, the process variable signal 112is oscillating due to stiction. Upon activation of the two-stepcompensation, the oscillations reduce or stop, and the valve 102 reachesits steady-state position. This is also shown in FIG. 7, where line 702represents the compensation signal 124, line 704 represents the controlsignal u provided to the valve 102 (the sum of output signal 116 andcompensation signal 124), and line 706 represents the position of thestem 202/plug 204 in the valve 102. It appears here that the use ofprocess or controller models may not be required, allowing this approachto be used in a wide variety of industrial settings.

FIG. 8 illustrates experimental results obtained using theabove-described technique. A water-flow system includes a linear needleplug valve assembly. The control valve represents a pneumatic air valve,and the actuator is configured “Air to Close” with a fail safe settingas “Open Fully.” No valve positioner is used. The control valveinitially shows negligible stiction (<0.2%). Stiction of about 5% of thecontroller span (0% to 100%) is introduced by tightening the stempacking, which commonly occurs in industrial settings. A linear variabledifferential transformer (LVDT) is used to measure stem position of thevalve. The results are shown in FIG. 8, where line 802 represents theprocess variable signal 112, line 804 represents the setpoint signal114, line 806 represents the compensation signal 124, and line 808represents the stem position in the valve.

In this example, the process variable signal 112 is oscillating due tostiction until activation of the stiction compensator 122 at time t=1200seconds. The oscillations are then reduced or eliminated and the stemmoves to its steady-state position with two compensating moves. Afterthe process variable signal 112 steadies at or near its setpoint, thestiction compensator 122 is switched off at time t=2000 seconds. Betweentimes t=1200 and t=2000, the stiction compensator 122 prevents the valvefrom moving away from its steady-state position. After time t=2000, adisturbance causes the valve to move away from its setpoint, and theoscillations caused by stiction reappear. This activates the stictioncompensator 122 at time t=2750 seconds, which again eliminates theoscillations and brings the valve to its steady-state position.

As shown in these examples, the stiction compensator 122 may beeffective in (i) moving a valve 102 to its steady-state positionrelatively quickly and (ii) maintaining the valve 102 at itssteady-state position. The examples shown in FIGS. 6 through 8 areprovided merely to illustrate example operations of the stictioncompensator 122. The stiction compensator 122 may operate in any otheror additional manner to move a valve 102 to its steady-state positionand/or to maintain the valve 102 at its steady-state position.

Although FIGS. 1 through 8 illustrate various embodiments and operationsof a process control system 100 and a stiction compensator 122, variouschanges may be made to FIGS. 1 through 8. For example, the processcontrol system 100 could include any number of valves 102, pipes 104,controllers 108, and stiction compensators 122. Also, the stictioncompensator 122 could be used in any other suitable system and could beused with any number of controllers 108 and/or valves 102. Further,while shown as a series of steps, various steps in the method 500 ofFIG. 5 could overlap or occur in parallel.

FIGS. 9 through 10B illustrate an example technique for qualitativelydetermining an amount of stiction in a valve 102. This may be useful,for example, during step 502 of FIG. 5. As noted above, to determine aqualitative stiction severity measure, a qualitative pattern recognitionapproach can be used to identify the patterns (such assquare/rectangular, triangular, or saw-toothed) that routinely occurwhen stiction is present.

Stiction often leads to the creation of clearly distinguishable patternsin various signals (such as process variable signal 112 and outputsignal 116). However, automatically recognizing and characterizing thesepatterns is often difficult due to many factors, including the fact thatstiction patterns are often asymmetric, non-linear, and time varying inboth frequency and magnitude.

A technique for qualitatively determining an amount of stiction in avalve 102 may involve accurately characterizing the oscillations in asignal. Many current techniques lack or fail to provide timelocalization of the oscillations, or characterizations of the starttimes of the oscillations and their consecutive zero-crossings. Withavailable information often limited to routine operating data (signals112-116), controller settings (such as Proportional, Integral,Derivative or “PID” settings), and loop type (such as flow, pressure,level, or temperature), time localization of an oscillating signal maybe very useful in identifying stiction.

As shown in FIG. 9, a method 900 for characterizing the oscillations ina signal involves three basic steps. The first step 902 removes anon-constant mean from the signal being analyzed (such as signal 112 or116). In some embodiments, this may involve the use of an adaptivemean-shifting procedure. In particular embodiments, the mean-shiftingprocedure may involve finding local maxima and minima points in thesignal being analyzed. Once the extreme points are identified, a cubicspline as an upper envelope can connect all of the local maxima, and alower envelope can be produced from the local minima. The mean of thedata can be computed by averaging the upper and lower envelopes. Theshifted signal may be obtained by subtracting the calculated mean fromthe actual signal.

The second step 904 involves computing the area of the mean shiftedsignal and normalizing the area. This step may help to reduce or removespurious zero-crossings that occur due to noise.

The third step 906 involves finding points at which slope changes occurin the area curve, which helps to identify the zero-crossings of thesignal being analyzed. In some embodiments, a clustering technique canbe employed for highly noisy data to group nearby zero-crossing points.An auto-correlation function test can further confirm the presence ofoscillation between the identified zero-crossings. Quiet periods (suchas time periods where oscillations are not present) in an intermittentlyoscillating signal can be identified using a qualitative trend approach.In addition, a time vector at which zero-crossings occur in the signalbeing analyzed can be generated.

This method 900 can be completely automated as it involves parametersthat are calculated from the signal being analyzed. This method 900 canalso operate on univariate data series. As described below, this method900 can be used when analyzing the signals 112-116 to determine aqualitative stiction severity measure.

A method 1000 for qualitatively determining an amount of stiction in avalve 102 is shown in FIGS. 10A and 10B. In general, the method 1000operates using sampled data (such as samples of signals 112-116). It isassumed that the sampling interval for data collection is adequate topreserve any relevant oscillating patterns. Typical sampling intervalsfor flow, pressure, level, and temperature loops can be one, five,thirty, and sixty seconds, respectively.

A portion of the data to be analyzed is received at step 1002. This mayinclude receiving data samples of the process variable signal 112, thesetpoint signal 114, and the output signal 116. At least some of thisdata can be mean-shifted as described above with respect to step 902 ofFIG. 9.

The presence of oscillations is determined at step 1004. Theoscillations could be detected, for example, using an auto-correlationtest. If no oscillations are detected, the method 1000 returns to step1002 to analyze additional data. Otherwise, the method 1000 continues tostep 1006, where the oscillations in the process variable signal 112(PV) and the output signal 116 (OP) are time characterized. This couldoccur as described above with respect to steps 904-906 of FIG. 9. Insome embodiments, a threshold number of oscillation cycles areidentified for analysis if possible, such as at least ten cycles. Inparticular embodiments, since the signals 112 and 116 oscillate in asimilar time range, the time stamps for the zero-crossings of one (suchas signal 116) can be used as the time stamps for the zero-crossings ofthe other (such as signal 112).

Using the input data and the time characterizations, test patterns forthe process variable signal 112 and the output signal 116 are generatedat step 1008. This could include, for example, selecting a portion ofeach signal 112 and 116 that contains a single full cycle of theoscillations (based on the identified zero-crossings). This could alsoinclude identifying the peak positive and negative amplitudes and theircorresponding time periods.

The process variable signal's test pattern is selected at step 1010. Atstep 1012, a reference template is generated for the selected testpattern. The reference template may include multiple patterns, such assquare/rectangular, triangular, trapezoidal, and sinusoidal patterns.The patterns may have the appropriate amplitude and frequency for theselected test pattern, which could be determined based on the identifiedpeak amplitudes and corresponding time periods.

A first template pattern is selected at step 1014. A pattern matching isperformed using the selected test pattern and the selected templatepattern at step 1016. This could include using Dynamic Time Warping toperform the pattern matching. A similarity measure (or a dissimilaritymeasure) of the two patterns is determined and stored at step 1018. Ifmore template patterns remain to be processed at step 1020, the method1000 returns to step 1014. Otherwise, the pattern from the template thatmost closely matches the selected test pattern is selected at step 1022.This could include identifying the template pattern having the highestsimilarity measure or the lowest dissimilarity measure.

At this point, a determination is made as to whether both the processvariable signal's test pattern and the output signal's test pattern havebeen analyzed at step 1024. If not, the output signal's test pattern isselected at step 1026. The output signal's test pattern could have thesame time range as the process variable signal's test pattern. Themethod 1000 then returns to step 1012, where the method 1000 identifiesa pattern in a reference template that most closely matches the outputsignal's test pattern.

After both test patterns have been analyzed, a determination is made asto whether stiction is detected at step 1028. For example, if none ofthe template patterns closely matches the test pattern for either signal112 or 116, this could result in a “no stiction” determination. If eachtest pattern closely matches one of its corresponding template patterns,the following table can be used to determine if stiction is present.

TABLE 1 Fast Process (Flow) Slow Process Integrating Dominant Dominant(Temperature, Process Level With Signal (I) Action (P) Action Pressure)(Level) PI Control OP Triangular Rectangular Triangular TriangularTriangular (Sharp) (Smooth) (Sharp) (Sharp) PV Square RectangularSinusoidal Triangular Parabolic/ (Sharp) TrapezoidalHere, the top row identifies the different types of control or processloops (flow, temperature, pressure, integrating or level, and level withPI control). If (for a specified type of loop) the identified templateshapes for the PV and OP test patterns match the patterns specified inTable 1, the stiction controller 122 may conclude that stiction ispresent in the valve 102. If (for a specified type of loop) one or moreof the identified template shapes for the PV and OP test patterns do notmatch the patterns specified in Table 1, the stiction controller 122 mayconclude that stiction is not present in the valve 102. For example, ifa temperature loop has an OP test pattern that is smooth triangular inshape and a PV test pattern that is sinusoidal in shape, this mayindicate stiction is present in the valve 102. If the temperature loophas an OP test pattern that is smooth triangular in shape and a PV testpattern that is square in shape, this may indicate stiction is notpresent in the valve 102.

If stiction is identified at step 1028, this particular oscillationcycle is marked as being “sticky” (meaning the oscillation cycle isindicative of stiction in the valve 102) at step 1030. If stiction isnot identified at step 1028, this particular oscillation cycle is notmarked as being “sticky.” This process may be repeated for eachindividual oscillation cycle identified in the signals being analyzed,such as for each of at least ten oscillation cycles. If additionaloscillation cycles remain to be processed at step 1032, the nextoscillation cycle is selected at step 1034, and the process returns tostep 1008 to analyze the next oscillation cycle.

If all of the oscillation cycles have been analyzed, a measure ofstiction is determined at step 1036. For example, the stiction measurecould be defined as the ratio of the number of cycles identified asbeing “sticky” versus the total number of cycles analyzed.

Although FIGS. 9 through 10B illustrate one example of a technique forqualitatively determining an amount of stiction in a valve 102, variouschanges may be made to FIGS. 9 through 10B. For example, while shown asa series of steps, some steps shown in the figures could overlap oroccur in parallel. As a particular example, steps 1008-1010 could occurin parallel with steps 1012-1014.

FIGS. 11A and 11B illustrate an example technique for quantitativelydetermining an amount of stiction in a valve 102. This may be useful,for example, during step 502 of FIG. 5. As noted above, to determine aquantitative stiction severity measure, a Hammerstein modelidentification procedure can be used to estimate the stiction parameterd in Equation (1). In general, the method 1100 operates using sampleddata (such as samples of signals 112-116). Here, the stictionidentification task may involve estimating the best process model (wherenot available) and the stiction band d simultaneously, such that theoperating data is modeled adequately.

As shown in FIG. 11A, a portion of the data to be analyzed is receivedat step 1102. This may include receiving data samples of the processvariable signal 112, the setpoint signal 114, and the output signal 116.At least some of this data can be denoised, such as by using a waveletdenoising technique. The data can also be normalized.

The presence of oscillations is determined at step 1104. Theoscillations could be detected, for example, using an auto-correlationtest. If no oscillations are detected, the method 1100 returns to step1102 to analyze additional data.

Otherwise, the method 1100 continues to step 1106, where a grid isgenerated for the stiction band d. If the process variable signal 112and output signal 116 are normalized, the stiction band d may generallylie in the range 0-1 or 0%-100%. In particular embodiments, aone-dimensional grid (with a range of 0-1) is generated for the stictionband d, with values separated by any suitable interval (such as 0.1). Avalue of d is then selected from the grid at step 1108.

A model is then selected for use with the selected value of d. Adecision is made as to whether a linear model of the process system isavailable at step 1110. The linear model could, for example, represent aprocess model (such as plant 406). If a linear model is available, thelinear model is selected at step 1112.

Otherwise, an Auto-Regressive Moving Average with exogenous variable(ARMAX) model is identified and analyzed at step 1114. In particularembodiments, the ARMAX model can be defined as follows:

$\begin{matrix}{{y(t)} = {{\frac{B\left( q^{- 1} \right)}{A\left( q^{- 1} \right)}{x\left( {t - 1} \right)}} + {\frac{C\left( q^{- 1} \right)}{A\left( q^{- 1} \right)}{e(t)}}}} & (15)\end{matrix}$where polynomials A(q⁻¹), B(q⁻¹), and C(q⁻¹) can be defined as:A(q ⁻¹)=1−a ₁ q ⁻¹ −a ₂ q ⁻² − . . . −a _(n) q ^(−n)  (16)B(q ⁻¹)=1+b ₁ q ⁻¹ +b ₂ q ⁻² + . . . +b _(m) q ^(−m)  (17)C(q ⁻¹)=1+c ₁ q ⁻¹ +c ₂ q ⁻² + . . . +c _(p) q ^(−p)  (18)and C(q⁻¹)/A(q⁻¹) represents model disturbances. The model can beanalyzed in any suitable manner, such as by using a loss function basedon the Akaike information criterion (AIC).

Via steps 1116-1118, the ARMAX model (with different model orders) isanalyzed, and the loss function can be determined for each model order.The ARMAX model having the lowest AIC value is then selected at step1120.

The selected model and the selected value of d are applied to the databeing analyzed at step 1122. This may include applying the selectedmodel and the selected value of d to the process variable signal 112 inorder to predict the output signal 116. The predicted output signal 116is then compared to the actual output signal 116 to determine a meansquare error at step 1124. If any grid points remain to be processed atstep 1126, the process returns to step 1108 to select and analyzeanother grid point.

Otherwise, all grid points have been analyzed, and the grid point withthe lowest mean square error is selected at step 1128. This grid pointmay represent the best estimate the stiction parameter d in Equation(1). An amount of stiction based on the selected grid point is thenreported at step 1130.

Although FIGS. 11A and 11B illustrate one example of a technique forquantitatively determining an amount of stiction in a valve 102, variouschanges may be made to FIGS. 11A and 11B. For example, while shown as aseries of steps, some steps shown in the figures could overlap or occurin parallel. Also, while a grid search has been illustrated in thefigures, other algorithms may be applied to find the optimum or selectedstiction band d.

In addition, fixing the stiction band d grid size (such as to values of0-1) and the band's interval or resolution may impact stictiondiagnosis. For example, after a certain value of the stiction band d,the mean square error may change marginally, which is explained usingthe stiction model in Equation (1). When the loop is oscillating, it canbe seen that, for any increase in the stiction band d beyond a valuethat is more than the actual stiction band, the position of the stem202/plug 204 remains unchanged as it retains its previous value. Sincethe input remains unchanged above a certain value from the true value ofd, the identified linear model remains the same. This can be used toreduce the grid search space considerably, thereby improving thecomputational efficiency. The resolution of the grid can also be fixedbased on hardware considerations of the control system. As an example,the controller output resolution may be limited by a digital-to-analog(D/A) output converter, which is often part of a distributed controlsystem (DCS). The resolution of the grid for the stiction band d canthus be set to the D/A resolution of the DCS system.

In some embodiments, various functions described above are implementedor supported by a computer program that is formed from computer readableprogram code and that is embodied in a computer readable medium. Thephrase “computer readable program code” includes any type of computercode, including source code, object code, and executable code. Thephrase “computer readable medium” includes any type of medium capable ofbeing accessed by a computer, such as read only memory (ROM), randomaccess memory (RAM), a hard disk drive, a compact disc (CD), a digitalvideo disc (DVD), or any other type of memory.

It may be advantageous to set forth definitions of certain words andphrases used throughout this patent document. The term “couple” and itsderivatives refer to any direct or indirect communication between two ormore elements, whether or not those elements are in physical contactwith one another. The terms “application” and “program” refer to one ormore computer programs, software components, sets of instructions,procedures, functions, objects, classes, instances, related data, or aportion thereof adapted for implementation in a suitable computer code(including source code, object code, or executable code). The terms“transmit” and “receive,” as well as derivatives thereof, encompass bothdirect and indirect communication. The terms “include” and “comprise,”as well as derivatives thereof, mean inclusion without limitation. Theterm “or” is inclusive, meaning and/or. The phrases “associated with”and “associated therewith,” as well as derivatives thereof, may mean toinclude, be included within, interconnect with, contain, be containedwithin, connect to or with, couple to or with, be communicable with,cooperate with, interleave, juxtapose, be proximate to, be bound to orwith, have, have a property of, or the like. The term “controller” meansany device, system, or part thereof that controls at least oneoperation. A controller may be implemented in hardware, firmware,software, or some combination of at least two of the same. Thefunctionality associated with any particular controller may becentralized or distributed, whether locally or remotely.

While this disclosure has described certain embodiments and generallyassociated methods, alterations and permutations of these embodimentsand methods will be apparent to those skilled in the art. Accordingly,the above description of example embodiments does not define orconstrain this disclosure. Other changes, substitutions, and alterationsare also possible without departing from the spirit and scope of thisdisclosure, as defined by the following claims.

1. An apparatus, comprising: at least one memory operable to storeoperating data associated with a valve; and at least one processoroperable to: identify whether the valve is suffering from stiction usingat least part of the operating data; and adjust a control signalproduced by a controller for the valve in order to at least partiallycompensate for the stiction suffered by the valve; wherein the at leastone processor is operable to adjust the control signal by (i) adjustingthe control signal to cause the valve to move into a steady-stateposition and (ii) adjusting the control signal to cause the valve toremain in the steady-state position; wherein the at least one processoris operable to adjust the control signal to cause the valve to move intothe steady-state position by (i) generating one or more firstcompensating signals for adjusting the control signal to cause the valveto move from a starting position into one or more new positions, (ii)predicting a next value of the control signal using a model of thecontroller, and (iii) generating, based on the predicted next value ofthe control signal, a second compensating signal for adjusting thecontrol signal to cause the valve to move from a last of the one or morenew positions into the steady-state position; wherein the at least oneprocessor is operable to identify whether the valve is suffering fromstiction by (i) determining a qualitative stiction severity measure byperforming qualitative pattern recognition and (ii) determining aquantitative stiction severity measure by performing a modelidentification procedure; and wherein the at least one processor isoperable to determine the qualitative stiction severity measure by:removing a non-constant mean from the at least part of the operatingdata to produce a mean-shifted signal; computing an area associated withthe mean-shifted signal and normalizing the area to produce an areacurve; and finding one or more points at which slope changes occur inthe area curve to identify one or more zero-crossings.
 2. The apparatusof claim 1, wherein the at least one processor is operable to generateone first compensating signal as defined by:(f _(k))_(t) =|m _(t) |+d where (f_(k))_(t) represents the firstcompensating signal at time t, m, represents a value of the controlsignal at time t, and d represents a value of a stiction band in astiction model.
 3. The apparatus of claim 2, wherein the firstcompensating signal is added to or subtracted from the control signal.4. The apparatus of claim 2, wherein the at least one processor isoperable to generate the second compensating signal as defined by:(f _(k))_(t+1) =−m _(t+1) where (f_(k))_(t+1) represents the secondcompensating signal at time t+1, and m_(t+1) represents the predictednext value of the control signal at time t+1.
 5. The apparatus of claim1, wherein the at least one processor is operable to adjust the controlsignal to cause the valve to remain in the steady-state position bygenerating a third compensating signal that satisfies a condition:|m _(t)+(f _(k))_(t) |<d where (f_(k))_(t) represents the thirdcompensating signal at time t, m_(t) represents a value of the controlsignal at time t, and d represents a value of a stiction band in astiction model.
 6. The apparatus of claim 1, wherein the at least oneprocessor is operable to adjust the control signal to cause the valve toremain in the steady-state position until a change in a setpoint occurs,the setpoint associated with the steady-state position of the valve. 7.A method, comprising: identifying whether a valve is suffering fromstiction; and adjusting a control signal produced by a controller forthe valve in order to at least partially compensate for the stictionsuffered by the valve; wherein adjusting the control signal comprises(i) adjusting the control signal to cause the valve to move into asteady-state position and (ii) adjusting the control signal to cause thevalve to remain in the steady-state position; wherein adjusting thecontrol signal to cause the valve to move into the steady-state positioncomprises (i) generating one or more first compensating signals foradjusting the control signal to cause the valve to move from a startingposition into one or more new positions, (ii) predicting a next value ofthe control signal using a model of the controller, and (iii)generating, based on the predicted next value of the control signal, asecond compensating signal for adjusting the control signal to cause thevalve to move from a last of the one or more new positions into thesteady-state position; wherein identifying whether the valve issuffering from stiction comprises (i) determining a qualitative stictionseverity measure by performing qualitative pattern recognition and (ii)determining a quantitative stiction severity measure by performing amodel identification procedure; and wherein determining the qualitativestiction severity measure comprises: detecting multiple oscillations inthe at least part of the operating data; time-characterizing thedetected oscillations; for each time-characterized oscillation,generating a test pattern associated with the oscillation, generating areference template comprising multiple patterns, and determining thatthe oscillation represents a “sticky” oscillation if the test patternmatches a specified one of the patterns in the reference template; anddetermining a measure of stiction based on a number of “sticky”oscillations.
 8. The method of claim 7, wherein generating the one ormore first compensating signals comprises generating one firstcompensating signal as defined by:(f _(k))_(t) =|m _(t) |+d where (f_(k))_(t) represents the firstcompensating signal at time t, m_(t) represents a value of the controlsignal at time t, and d represents a value of a stiction band in astiction model.
 9. The method of claim 8, wherein generating the secondcompensating signal comprises generating one second compensating signalas defined by:(f _(k))_(t+1) =−m _(t+1) where (f_(k))_(t+1) represents the secondcompensating signal at time t+1, and m_(t+1) represents the predictednext value of the control signal at time t+1.
 10. The method of claim 7,wherein adjusting the control signal to cause the valve to remain in thesteady-state position comprises generating a third compensating signalthat satisfies a condition:|m _(t)+(f _(k))_(t) |<d where (f_(k))_(t) represents the thirdcompensating signal at time t, m_(t) represents a value of the controlsignal at time t, and d represents a value of a stiction band in astiction model.
 11. The method of claim 7, wherein adjusting the controlsignal to cause the valve to remain in the steady-state positioncontinues until a change in a setpoint occurs, the setpoint associatedwith the steady-state position of the valve.
 12. The method of claim 7,wherein the controller is operable to use measurement data from a sensorto generate the control signal.
 13. The method of claim 7, whereinidentifying whether the valve is suffering from stiction comprisesdetermining whether the valve is suffering from a threshold amount ofstiction.
 14. A computer readable medium embodying a computer program,the computer program comprising: computer readable program code foridentifying whether a valve is suffering from stiction; and computerreadable program code for adjusting a control signal produced by acontroller for the valve in order to at least partially compensate forthe stiction suffered by the valve; wherein the computer readableprogram code for adjusting the control signal comprises (i) computerreadable program code for adjusting the control signal to cause thevalve to move into a steady-state position and (ii) computer readableprogram code for adjusting the control signal to cause the valve toremain in the steady-state position; wherein the computer readableprogram code for adjusting the control signal to cause the valve to moveinto the steady-state position comprises (i) computer readable programcode for generating one or more first compensating signals for adjustingthe control signal to cause the valve to move from a starting positioninto one or more new positions, (ii) computer readable program code forpredicting a next value of the control signal using a model of thecontroller, and (iii) computer readable program code for generating,based on the predicted next value of the control signal, a secondcompensating signal for adjusting the control signal to cause the valveto move from a last of the one or more new positions into thesteady-state position; wherein the computer readable program code foridentifying whether the valve is suffering from stiction comprisescomputer readable program code for (i) determining a qualitativestiction severity measure by performing qualitative pattern recognitionand (ii) determining a quantitative stiction severity measure byperforming a model identification procedure; and wherein the computerreadable program code for determining the quantitative stiction severitymeasure comprises computer readable program code for: generating a gridcomprising multiple values of a stiction band; for each value in thegrid, selecting that value of the stiction band, selecting a model foruse with that value of the stiction band, applying the selected value ofthe stiction band and the selected model to the at least part of theoperating data, and determining an error associated with the applicationof the selected value of the stiction band and the selected model to theat least part of the operating data; identifying the lowest error; anddetermining a measure of stiction based on the selected value of thestiction band and the selected model that are associated with the lowesterror.
 15. The computer readable medium of claim 14, wherein thecomputer readable program code for generating the one or more firstcompensating signals comprises computer readable program code forgenerating one first compensating signal as defined by:(f _(k))_(t) =|m _(t) +d where (f_(k))_(t) represents the firstcompensating signal at time t, m_(t) represents a value of the controlsignal at time t, and d represents a value of a stiction band in astiction model.
 16. The computer readable medium of claim 15, whereinthe computer readable program code for generating the secondcompensating signal comprises computer readable program code forgenerating one second compensating signal as defined by:(f _(k))_(t+1) =−m _(t+1) where (f_(k))_(t+1) represents the secondcompensating signal at time t+1, and m_(t+1) represents the predictednext value of the control signal at time t+1.
 17. The computer readablemedium of claim 14, wherein the computer readable program code foradjusting the control signal to cause the valve to remain in thesteady-state position comprises computer readable program code forgenerating a third compensating signal that satisfies a condition:|m _(t)+(f _(k))_(t) |<d where (f_(k))_(t) represents the thirdcompensating signal at time t, m_(t) represents a value of the controlsignal at time t, and d represents a value of a stiction band in astiction model.
 18. The apparatus of claim 1, wherein the at least oneprocessor is operable to determine the qualitative stiction severitymeasure by: detecting multiple oscillations in the at least part of theoperating data; time-characterizing the detected oscillations; for eachtime-characterized oscillation, generating a test pattern associatedwith the oscillation, generating a reference template comprisingmultiple patterns, and determining that the oscillation represents a“sticky” oscillation if the test pattern matches a specified one of thepatterns in the reference template; and determining a measure ofstiction based on a number of “sticky” oscillations.
 19. The apparatusof claim 1, wherein the at least one processor is operable to determinethe quantitative stiction severity measure by performing a Hammersteinmodel identification procedure to estimate a stiction parameter d in:$x_{t} = \left\{ \begin{matrix}x_{t - 1} & {{{if}\mspace{14mu}{{u_{t} - x_{t - 1}}}} \leq d} \\u_{t} & {otherwise}\end{matrix} \right.$ where x_(t) and x_(t−1) represent present and pastmovements associated with the valve and μ_(t) represents a present valueof the control signal.
 20. The apparatus of claim 1, wherein the atleast one processor is operable to determine the quantitative stictionseverity measure by: generating a grid comprising multiple values of astiction band; for each value in the grid, selecting that value of thestiction band, selecting a model for use with that value of the stictionband, applying the selected value of the stiction band and the selectedmodel to the at least part of the operating data, and determining anerror associated with the application of the selected value of thestiction band and the selected model to the at least part of theoperating data; identifying the lowest error; and determining a measureof stiction based on the selected value of the stiction band and theselected model that are associated with the lowest error.
 21. Theapparatus of claim 1, wherein: the at least one processor is operable togenerate the one or more first compensating signals by selecting anyfirst compensation that causes the valve to move from a stuck positionwithout saturating; and the at least one processor is operable togenerate the second compensating signal by selecting a secondcompensation that is not dependent on the first compensation.